This longstanding anomaly motivates a lot of continuous theoretical explanations with variants of new physics models.
For comprehensive reviews, see Refs. [1]}, [2]}. We are particularly interested in two possibilities that have been broadly considered to address this anomaly, with either a light gauge boson (or neutral scalar) interacting with muons.
In particular, we consider a dark U(1) model and a dark \(Z_3\)  scalar.
We note that for the explanation of the \((g-2)_\mu \)  anomaly with a light gauge boson of \(U(1)_{\mu -\tau }\) , the required gauge coupling is of order \(\mathcal {O}(10^{-4}-10^{-3})\)  for \(\mathcal {O}(1-10^2)\)  MeV light gauge boson and is too small to trigger a first-order phase transition (FOPT).
In the meantime, we notice that
the NANOGrav Collaboration reports a strong evidence of a stochastic common-spectrum process [3]} in the 12.5-yr data set.
Among various gravitational wave background explanations of the process, the phase transition occurring at low temperature of around \(T_\star \sim \mathcal {O}(1-10)\)  MeV can successfully fit the dataset [4]}, [5]}, [6]}, [7]}, [8]}. Therefore, it is of great interest to study the possible common new physics origin for the \(g-2\)  anomaly together with pulsar timing arrays experiments.
